introduction uniform block permutation n - dartmouth …orellana/blockperm.pdf · the hopf algebra...

THE HOPF ALGEBRA OF UNIFORM BLOCK PERMUTATIONS

MARCELO AGUIAR AND ROSA C. ORELLANA

Abstract. We introduce the Hopf algebra of uniform block permutations and show thatit is self-dual, free, and cofree. These results are closely related to the fact that uniformblock permutations form a factorizable inverse monoid. This Hopf algebra contains the Hopfalgebra of permutations of Malvenuto and Reutenauer and the Hopf algebra of symmetricfunctions in non-commuting variables of Gebhard, Rosas, and Sagan. These two embeddingscorrespond to the factorization of a uniform block permutation as a product of an invertibleelement and an idempotent one.

Introduction

A uniform block permutation of [n] is a certain type of bijection between two set partitionsof [n]. When the blocks of both partitions are singletons, a uniform block permutation issimply a permutation of [n]. Let Pn be the set of uniform block permutations of [n] and Sn

the subset of permutations of [n]. The set Pn is a monoid in which the invertible elementsare precisely the elements of Sn. These notions are reviewed in Section 1.

This paper introduces and studies a graded Hopf algebra based on the set of uniform blockpermutations of [n] for all n ≥ 0, by analogy with the graded Hopf algebra of permutationsof Malvenuto and Reutenauer [17].

Let V be a complex vector space. Classical Schur-Weyl duality states that the symmetricgroup algebra can be recovered from the diagonal action of GL(V ) on V ⊗n: if dim V ≥ nthen

(1) CSn∼= EndGL(V )(V

⊗n) .

Malvenuto and Reutenauer deduce from here the existence of a multiplication among per-mutations as follows. Given σ ∈ Sp and τ ∈ Sq, view them as linear endomorphisms of thetensor algebra

T (V ) :=⊕n≥0

V ⊗n

Date: October 30, 2006.2000 Mathematics Subject Classification. Primary: 05E99, 16W30; Secondary: 16G99, 20C30.Key words and phrases. Hopf algebra, factorizable inverse monoid, uniform block permutation, set parti-

tion, symmetric functions, Schur-Weyl duality.Aguiar supported in part by NSF grant DMS-0302423.Orellana supported in part by the Wilson Foundation.

1

2 MARCELO AGUIAR AND ROSA C. ORELLANA

by means of (1) (σ acts as 0 on V ⊗n if n 6= p, similarly for τ). The tensor algebra is a Hopfalgebra, so we can form the convolution product of any two linear endomorphisms:

T (V )∆−→ T (V )⊗ T (V )

σ⊗τ−−→ T (V )⊗ T (V )m−→ T (V ) ,

where ∆ and m are the coproduct and product of the tensor algebra. Since these two mapscommute with the action of GL(V ), the convolution of σ and τ belongs to EndGL(V )(V

⊗n),where n = p + q. Therefore, there exists an element σ ∗ τ ∈ CSn whose right action equalsthe convolution of σ and τ . This is the product of the algebra of permutations. It turns outthat a suitable coproduct can also be defined. The result is the Hopf algebra of permutationsof Malvenuto and Reutenauer.

The same argument can be applied to define a convolution product on the direct sum ofthe centralizer algebras EndG(V ⊗n), starting from a linear action of a group G on a vectorspace V . In this paper we consider one such instance in which in addition a compatiblecoproduct can also be defined.

Consider the complex reflection group G(r, 1, m). The monomial representation is a certainlinear action of this group on an m-dimensional space V (Section 2.1). A result of Tanabeidentifies the centralizer of V ⊗n with the monoid algebra of uniform block permutations (if rand m are big in comparison to n; see Proposition 2.1). The convolution product is thereforedefined on the space

⊕n CPn. An explicit description of this operation, similar to that for

the convolution product of permutations, is given in Section 2.2, along with the definition ofa compatible coproduct which turns this space into a graded Hopf algebra (Theorem 2.6).The Hopf algebra of permutations is a Hopf subalgebra (Proposition 2.7).

Except for the description of CPn as a centralizer algebra, it is not necessary to work overthe complex numbers. Accordingly, we work from the start over an arbitrary commutativering k. All modules considered will be free (having in fact a distinguished basis) and aretherefore referred to as spaces.

The Hopf algebra structure of uniform block permutations is studied in Section 3.1. Propo-sition 3.1 states that this Hopf algebra is self-dual. This result is related to the fact that Pn

carries an involution which turns it into an inverse monoid. This is moreover a factorizableinverse monoid, which refers to the fact that any element is the product of an invertibleelement and an idempotent element. This is used in Section 3.2 to define a partial order onPn analogous to the weak order on Sn. Following [2], a second linear basis of the space kPn

is defined in Section 3.3 by performing Mobius inversion with respect to this partial order.The convolution product admits a simple description on this basis (Proposition 3.4) fromwhich it follows that the algebra of uniform block permutations is free (and by self-dualityalso cofree) (Corollary 3.5).

It turns out that the Hopf algebra of uniform block permutations contains another distin-guished Hopf subalgebra, namely the Hopf algebra of symmetric functions in non-commutativevariables (Theorem 4.1). This algebra was introduced by Wolf [23]. It is based on the set ofset partitions of [n] for all n ≥ 0. Sagan brought it to the forefront in a series of papers withGebhard and Rosas [11, 12, 19]. Bergeron, Hohlweg, Rosas, and Zabrocki studied the Hopf

HOPF ALGEBRA OF UNIFORM BLOCK PERMUTATIONS 3

algebra structure in [4, 5]. Connections between this and other combinatorial Hopf algebrasare studied in [1]. The embedding of this Hopf algebra is in a sense complementary to theembedding of the algebra of permutations: permutations arise as the invertible elementsof the monoid of uniform block permutations, while set partitions arise as the idempotentelements therein. This is discussed in Section 4.

1. Uniform block permutations

1.1. Set partitions. Let n be a non-negative integer and let [n] := {1, 2, . . . , n}. A setpartition of [n] is a collection of non-empty disjoint subsets of [n], called blocks, whoseunion is [n]. For example, A =

{{2, 5, 7}{1, 3}{6, 8}{4}

}, is a set partition of [8] with 4

blocks. We often specify a set partition by listing the blocks from left to right so thatthe sequence formed by the minima of the blocks is increasing, and by listing the elementswithin each block in increasing order. For instance, the set partition above will be denotedA = {1, 3}{2, 5, 7}{4}{6, 8}. We use A ` [n] to indicate that A is a set partition of [n].

The type of a set partition A of [n] is the partition of n formed by the sizes of the blocks ofA. The symmetric group Sn acts on the set of set partitions of [n]: given σ ∈ Sn and A ` [n],σ(A) is the set partition whose blocks are σ(A) for A ∈ A. The orbit of A consists of thoseset partitions of the same type as A. The stabilizer of A consists of those permutations thatpreserve the blocks, or that permute blocks of the same size. Therefore, the number of setpartitions of type 1m12m2 . . . nmn (mi blocks of size i) is

(2)n!

m1! · · ·mn!(1!)m1 · · · (n!)mn.

1.2. The monoid of uniform block permutations. The monoid (and the monoid alge-bra) of uniform block permutations has been studied by FitzGerald [10] and Kosuda [14, 15]in analogy to the partition algebra of Jones and Martin [13, 18].

A block permutation of [n] consists of two set partitions A and B of [n] with the samenumber of blocks and a bijection f : A → B. For example, if n = 3, f({1, 3}) = {3} andf({2}) = {1, 2} then f is a block permutation. A block permutation is called uniform if itmaps each block of A to a block of B of the same cardinality. For example, f({1, 3}) = {1, 2},f({2}) = {3} is uniform. Each permutation may be viewed as a uniform block permutationfor which all blocks have cardinality 1. In this paper we only consider block permutationsthat are uniform.

To specify a uniform block permutation f : A → B we must choose two set partitions Aand B of the same type 1m1 . . . nmn and for each i a bijection between the set of blocks ofsize i of A and the set of blocks of size i of B. Since there are mi blocks of size i, it followsfrom (2) that the total number of uniform block permutations of [n] is

(3) un :=∑

1m1 ...nmn`n

(n!

(1!)m1 · · · (n!)mn

)21

m1! · · ·mn!


where the sum runs over all partitions of n. Starting at n = 0, the first values are

1, 1, 3, 16, 131, 1496, 22482, . . .

This is sequence A023998 in [21]. These numbers and generalizations are studied in [20]; inparticular, the following recursion is given in [20, equation (11)]:

un+1 =n∑

k=0

(n

k

)(n + 1

k

)uk , u0 = 1 .

We represent uniform block permutations by means of graphs. For instance, either one ofthe two graphs in Figure 1 represents the uniform block permutation f given by

{1, 3, 4} → {3, 5, 6}, {2} → {4}, {5, 7} → {1, 2}, {6} → {8}, and {8} → {7} .

rr"

""

""

""

"

1

1

rr

2

2

rr

��

��

3

3

rr

\\

\\

\

4

4

rr

��

5

5

rr

6

6

rr

7

7

rr

\\

\\

\

8

8

\\

\\

\

��

\\

\\

\

∼ rr"

""

""

""

"

1

1

rr

2

2

rr

��

��

3

3

rr

\\

\\

\

4

4

rr

5

5

rr

6

6

rr

7

7

rr

\\

\\

\

8

8

\\

\\

\

��

\\

\\

\

Figure 1. Two graphs representing the same uniform block permutation

Different graphs may represent the same uniform block permutation. For a graph torepresent a uniform block permutation f : A → B of [n] the vertex set must consist oftwo copies of [n] (top and bottom) and each connected component must contain the samenumber of vertices on the top as on the bottom. The set partition A is read off from theadjacencies on the top, B from those on the bottom, and f from those in between.

The diagram of f is the unique representing graph in which all connected components arecycles and the elements in each cycle are joined in order, as in the second graph of Figure 1.

The set Pn of block permutations of [n] is a monoid. The product g · f of two uniformblock permutations f and g of [n] is obtained by gluing the bottom of a graph representingf to the top of a graph representing g. The resulting graph represents a uniform blockpermutation which does not depend on the graphs chosen. An example is given in Figure 2.Note that gluing the diagram of f to the diagram of g may not result in the diagram of g · f .

The identity is the uniform block permutation that maps {i} to {i} for all i. Viewingpermutations as uniform block permutations as above, we get that the symmetric group Sn

is a submonoid of Pn.We recall a presentation of the monoid Pn given in [10, 14, 15]. Consider the uniform

block permutations bi and si with diagrams


f =

rr��rr@

@�

�rr @@

rr rr��rr@

@g =

rr��rr rr��rrPPPPPP rr@

@rr@

@

g · f =

rr��rr@

@�

�rr @@

rr rr��rr@

@

rr��rr rr��rrPPPPPP rr@

@rr@

@

=rr rr��

��rr��rrHH

HH rr rr

Figure 2. Product of uniform block permutations

bi =

rr

1

1

· · ·rr

i− 1

i− 1

rri

i

rr

i + 1

i + 1

rri + 2

i + 2

· · ·rrn

n

si =

rr

1

1

· · ·rr

i− 1

i− 1

rr��

��

i

i

rrLL

LL

i + 1

i + 1

rri + 2

i + 2

· · ·rrn

n

The monoid Pn is generated by the elements {bi, si | 1 ≤ i ≤ n−1} subject to the followingrelations:

(1) s2i = 1, b2

i = bi, 1 ≤ i ≤ n− 1;(2) sisi+1si = si+1sisi+1, sibi+1si = si+1bisi+1, 1 ≤ i ≤ n− 2;(3) sisj = sjsi, bisj = sjbi, |i− j| > 1;(4) bisi = sibi = bi, 1 ≤ i ≤ n− 1;(5) bibj = bjbi, 1 ≤ i, j ≤ n− 1.

The submonoid generated by the elements si, 1 ≤ i ≤ n − 1 is the symmetric group Sn,viewed as a submonoid of Pn as above.

We will see in Sections 3.1 and 3.2 that Pn is a factorizable inverse monoid. Therefore, apresentation for Pn may also be derived from the results of [9].

1.3. An ideal indexed by set partitions. Let kPn be the monoid algebra of Pn over acommutative ring k.

Given a set partition A ` [n], let ZA ∈ kPn denote the sum of all uniform block permuta-tions f : A → B, where B varies:

(4) ZA :=∑

f :A→B

f .

For instance,


Z{1,3}{2,4} = r r r rr r r r��A

A + r r r rr r r r+ r r r rr r r r

��A

A + r r r rr r r r��A

A + r r r rr r r r��A

A��A

A + r r r rr r r r��

�@

@A

A

Given σ ∈ Sn and A ` [n], the set partition σ(A) ` [n] was defined in Section 1.1.Given i ∈ [n], let Ai denote the set partition obtained by merging the blocks of i and i+1

of A and keeping the other blocks of A unaltered.

Proposition 1.1. Let A be a set partition of [n] and σ a permutation of [n]. Then

σ · ZA = ZA and ZA · σ = Zσ−1(A) .

In addition,

ZA · bi =

{ZA if i and i + 1 belong to the same block of A(|A|+|A′|

|A|

)ZAi

if i and i + 1 belong to different blocks A and A′ of A.

Proof. Let f : A → B be a summand of ZA. Then σ ·f : A → σ(B). Thus left multiplicationby σ preserves the set of uniform block permutations with domain A. Since σ is invertible,this is a bijection. Therefore, σ ·ZA = ZA. Similarly, right multiplication by σ is a bijectionfrom the set of uniform block permutations with domain A to the set of uniform blockpermutations with domain σ−1(A). Hence ZA · σ = Zσ−1(A).

Let A and A′ be the blocks of i and i + 1 in the set partition A (they may coincide).Multiplying the uniform block permutation f on the right by bi has the effect of connectingthe vertices i and i + 1 in A. The domain of f · bi is therefore the set partition Ai andthe codomain is the set partition B obtained by merging the blocks f(A) and f(A′) of B.Moreover, f · bi : Ai → B is the uniform block permutation such that (f · bi)(A ∪ A′) =f(A) ∪ f(A′) and (f · bi)(A

′′) = f(A′′) on every other block A′′ of A.On the other hand, let g : Ai → C be a uniform block permutation. The block g(A ∪ A′)

of C can be decomposed as the disjoint union of two subsets C and C ′ with |C| = |A| and

|C ′| = |A′| in(|A+A′|

|A|

)ways. For each such decomposition there is a unique set partition

B and a unique uniform block permutation f : A → B such that B = C and f · bi = g.Therefore,

ZA · bi =

(|A + A′||A|

)ZAi

.

�

Let Zn denote the subspace of kPn linearly spanned by the elements ZA as A runs overall set partitions of [n].

Corollary 1.2. Zn is a right ideal of the monoid algebra kPn.

2. The Hopf algebra of uniform block permutations

In this section we define the Hopf algebra of uniform block permutations. It contains theHopf algebra of permutations of Malvenuto and Reutenauer as a Hopf subalgebra.


2.1. Schur-Weyl duality for uniform block permutations. Let r and m be positiveintegers. Let Cr denote the cyclic group of order r with generator t:

Cr := 〈t | tr = 1〉.Consider the complex reflection group

G(r, 1, m) := Cr o Sm .

Let V be the monomial representation of G(r, 1, m). Thus, V is an m-dimensional complexvector space with a basis {e1, e2, . . . , em} on which G(r, 1, m) acts as follows:

t · e1 = e2πi/re1 , t · ei = ei for i > 1, and σ · ei = eσ(i) for σ ∈ Sm.

Consider now the diagonal action of G(r, 1, m) on the tensor powers V ⊗n,

g · (ei1ei2 · · · ein) = (g · ei1)(g · ei2) · · · (g · ein) .

The centralizer of this representation has been calculated by Tanabe.Proposition 2.1. [22] There is a right action of the monoid Pn on V ⊗n determined by

(ei1 · · · ein) · bj = δ(ij, ij+1)ei1 · · · ein and (ei1 · · · ein) · σ = eiσ(1)· · · eiσ(n)

for 1 ≤ i ≤ n − 1 and σ ∈ Sn. This action commutes with the left action of G(r, 1, m) onV ⊗n. Moreover, if m ≥ 2n and r > n then the resulting map

(5) CPn → EndG(r,1,m)(V⊗n)

is an isomorphism of algebras.

As explained in the introduction, this result can be used to deduce the existence of aproduct on the space

P :=⊕n≥0

kPn

for which the mapP → End(T (V ))

(resulting from (5)) is a morphism of algebras, where End(T (V )) is viewed as an algebraunder the convolution product. We proceed to describe the product on P in explicit termsand to enlarge this structure to a graded connected Hopf algebra.

2.2. Product and coproduct of uniform block permutations. Let f and g be uniformblock permutations of [p] and [q] respectively. Adding p to every entry in the diagram of gand placing it to the right of the diagram of f we obtain the diagram of a uniform blockpermutation of [p + q], called the concatenation of f and g and denoted f × g. Figure 3shows an example.

Let Sh(p, q) denote the set of (p, q)-shuffles, that is, those permutations ξ ∈ Sp+q such that

ξ(1) < ξ(2) < · · · < ξ(p) and ξ(p + 1) < ξ(p + 2) < · · · < ξ(p + q) .

Let shp,q ∈ kSp+q denote the sum of all (p, q)-shuffles.


f =rr��

rr@@

rr@@

rr1 2 3 4

1 2 3 4

g =rr �

�rr�

�rr rr XXXXXXXX rr�

��1 2 3 4 5

1 2 3 4 5

f × g =rr��

��rr@@

rr@@

rr1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

rr ��

rr��

rr rr XXXXXXXX rr��

Figure 3. Concatenation of diagrams

The product ∗ on P is defined by

(6) f ∗ g := shp,q · (f × g) ∈ kPp+q

for all f ∈ Pp and g ∈ Pq, and extended by linearity. It is easy to see that this productcorresponds to convolution of endomorphisms of the tensor algebra via the map (5), whenk = C.

For example,rr rr ∗ rr��rr@@ =

rr rr rr rr��@@ +rr rr rr rr@@��

�@@ +

rr rr rr rrHHH�� +

rr rr rr@@rr@@��

@@ +rr rr@@

rr rrHHH��

+

rr rr rrHHH rrHHH��

��

The set P0 consists of the unique uniform block permutation of [0]. It is represented bythe empty diagram, which we denote by ∅. It is the unit element for the product ∗.

A breaking point of a set partition B ` [n] is an integer i ∈ {0, 1, . . . , n} for which thereexists a subset S ⊆ B such that

(7)⋃B∈S

B = {1, . . . , i} and (hence)⋃

B∈B\S

B = {i + 1, . . . , n} .

Note that i = 0 and i = n are breaking points of any set partition B.Given a uniform block permutation f : A → B, we say that i is a breaking point of f if

it is a breaking point of B, and we let B(f) denote the set of breaking points of f . If f is apermutation, that is if all blocks of f are of size 1, then B(f) = {0, 1, . . . , n}. In terms ofthe diagram of f , i ∈ B(f) if it is possible to put a vertical line between the first i and thelast n− i vertices in the bottom row without intersecting any edges joining bottom vertices.

f =rr��

rr��rrHHHH rr rr rr�

�rrPPPPPP rr ⇒ B(f) = {0, 1, 2, 6, 8}.


Lemma 2.2. If i is a breaking point of f , then there exists a unique (i, n− i)-shuffle ξ ∈ Sn

and unique uniform block permutations f(i) ∈ Pi and f ′(n−i) ∈ Pn−i such that

f = (f(i) × f ′(n−i)) · ξ−1 .

Conversely, if such a decomposition exists, i is a breaking point of f .

We illustrate this statement with an example where i = 4 and ξ =

(1 2 3 4 5 62 3 5 6 1 4

):

rrXXXXXXXXr

r�

� rr

�� r

rHHHHr

r�� r

r��

f(4) f ′(2)f =

= ξ−1

Proof. Suppose i is a breaking point of f : A → B. Let S be the subset of B as in (7). Write⋃B∈S

f−1(B) = {a1, a2, . . . , ai} and⋃

B∈B\S

f−1(B) = {ai+1, ai+2, . . . , an}

with 1 ≤ a1 < a2 < · · · < ai, ai+1 < ai+2 < · · · < an ≤ n. Define ξ ∈ Sh(i, n− i) by

ξ(r) = ar for r = 1, . . . , n.

By construction, any element of [i] is connected only to elements of [i] in the diagram of f ·ξ.Therefore, there exist f(i) ∈ Pi and f ′(n−i) ∈ Pn−i such that f · ξ = f(i) × f ′(n−i).

Assume that such a decomposition is given. Since i is a breaking point of f(i)× f ′(n−i) andthe codomain of f · ξ is the same as the codomain of f , we have that i is a breaking pointof f . Hence we must have ξ([i]) = {a1, a2, . . . , ai}, which makes the shuffle ξ unique. Thenf(i) and f ′(n−1) are determined from f · ξ = f(i) × f ′(n−i). �

We are now ready to define the coproduct on P . Given f ∈ Pn set

(8) ∆(f) :=∑

i∈B(f)

f(i) ⊗ f ′(n−i),

where f(i) and f ′(n−i) are as in Lemma 2.2. An example follows.

f = rr��rr rr@@ rr@@ rr rr��@@��

∆(f) = f ⊗ ∅ + rr��rr rr@@ ⊗ rr rr rr��@@ + rr��rr@

@ rr rr�� rr@@ ⊗ rr+ ∅ ⊗ f .


We define the counit ε : P → k by

ε(f) =

{1 if f = ∅ ∈ P0,

0 if f ∈ Pn, n ≥ 1.

Remark 2.3. Recall that an element x ∈ P is called primitive if ∆(x) = x ⊗ ∅ + ∅ ⊗ x.Every uniform block permutation with breaking set {0, n} is primitive, but there are otherprimitive elements in P . For example, the following element of kP3 is primitive:

rr rr rr��@@ − rr rr�� rrHHH��

The primitive elements of P are determined in Section 3.3.

For the proof of the next theorem we need the following element of the symmetric group.Given p, q ≥ 0, let ξp,q ∈ Sp+q be the permutation

(9) ξp,q :=

(1 2 . . . p p + 1 p + 2 . . . p + q

q + 1 q + 2 . . . q + p 1 2 . . . q

).

This is a (p, q)-shuffle (it is in fact the maximum element of Sh(p, q) under the weak order;see Section 3.2). The diagram of ξ3,4 is shown below.r

r��

��

�rr��

��

��

rr��

��

��

rr��

��

��

rrHH

HHHHHH

rrHH

HHHHHH

rrHH

HHHHHH

The inverse of ξp,q is ξq,p. Let 1n ∈ Sn denote the identity permutation. We need thefollowing familiar properties.

Lemma 2.4. Let p, q ≥ 0. For any f ∈ Pq, g ∈ Pq, we have

(10) ξp,q · (f × g) = (g × f) · ξp,q .

Lemma 2.5. Let a, b, c, d ≥ 0. For any shuffles ξ1 ∈ Sh(a, b) and ξ2 ∈ Sh(c, d), the permu-tation

(ξ1 × ξ2) · (1a × ξc,b × 1d)

is an (a + c, b + d)-shuffle. Conversely, let p, q, r, s ≥ 0 be such that

p + q = r + s.

Given a shuffle ξ ∈ Sh(p, q), there are unique numbers a, b, c, d ≥ 0 and unique shufflesξ1 ∈ Sh(a, b) and ξ2 ∈ Sh(c, d) such that

a + b = r, c + d = s, a + c = p, b + d = q,

and

ξ = (ξ1 × ξ2) · (1a × ξc,b × 1d).


Proof. The first assertion is straightforward. For the converse, given ξ ∈ Sh(p, q), we have

ξ−1({1, . . . , r}) = {1, . . . , a} ∪ {p + 1, . . . , p + b}for some a, b ≥ 0 such that a ≤ p, b ≤ q, and a + b = r (since ξ is increasing on {1, . . . , p}and on {p+1, . . . , p+ q}). Let c = p− a and d = q− b. A straightforward calculation showsthat

ξ · (1a × ξb,c × 1d) = ξ1 × ξ2

for some shuffles ξ1 ∈ Sh(a, b) and ξ2 ∈ Sh(c, d). This proves existence. The numbers aand b are uniquely determined from ξ−1({1, . . . , r}) as above, and then c, d and ξ1, ξ2 aredetermined as well. �

Theorem 2.6. The graded vector space P equipped with the product ∗, coproduct ∆, unit ∅,and counit ε, is a graded connected Hopf algebra.

Proof. Associativity and coassociativity follow from basic properties of shuffles (for the prod-uct one may also appeal to (5) and associativity of the convolution product). The existenceof the antipode is guaranteed in any graded connected bialgebra. The compatibility between∆ and ∗ requires a special argument.

Take f ∈ kPp and g ∈ kPq. On one hand we have

∆(f ∗ g)(6)=

∑ξ∈Sh(p,q)

∆(ξ · (f × g)

)(8)=

∑ξ∈Sh(p,q)

r∈B(ξ·(f×g))

α⊗ β,

here we have written, for each breaking point r ∈ B(ξ · (f × g)),

(11) ξ · (f × g) · η = α× β,

where α ∈ kPr, β ∈ kPp+q−r, and η ∈ Sh(r, p + q − r) are the unique elements afforded byLemma 2.2.

On the other hand, we have

∆(f) ∗∆(g) =∑

a∈B(f)b∈B(g)

(f ′(a) ∗ g′(b))⊗ (f ′′(p−a) ∗ g′′(q−b))

=∑

ξ1∈Sh(a,b)ξ2∈Sh(p−a,q−b)

∑a∈B(f)b∈B(g)

ξ1 · (f ′(a) × g′(b))⊗ ξ2 · (f ′′(p−a) × g′′(q−b)).

Here, for each pair of breaking points a ∈ B(f) and b ∈ B(g), we have written

(12) f · η1 = f ′(a) × f ′′(p−a) and g · η2 = g′(b) × g′′(q−b),

as in Lemma 2.2, with η1 ∈ Sh(a, p− a) and η2 ∈ Sh(b, q − b).We show that any summand in ∆(f) ∗∆(g) also occurs in ∆(f ∗ g) and viceversa.Given breaking points a, b and shuffles η1, η2 as in (12), define

η := (η1 × η2) · (1a × ξb,p−a × 1q−b).


In addition, given shuffles ξ1 ∈ Sh(a, b) and ξ2 ∈ Sh(p− a, q − b), define

ξ := (ξ1 × ξ2) · (1a × ξp−a,b × 1q−b).

Then η ∈ Sh(a + b, p + q − a− b) and ξ ∈ Sh(p, q), by Lemma 2.5. Moreover, we have:

ξ · (f × g) · η = (ξ1 × ξ2) · (1a × ξp−a,b × 1q−b) · (f × g) · (η1 × η2) · (1a × ξb,p−a × 1q−b)

= (ξ1 × ξ2) · (1a × ξp−a,b × 1q−b) · (f · η1)× (g · η2) · (1a × ξb,p−a × 1q−b)

(12)= (ξ1 × ξ2) · (1a × ξp−a,b × 1q−b) · (f ′(a) × f ′′(p−a) × g′(b) × g′′(q−b)) · (1a × ξb,p−a × 1q−b)

(10)= (ξ1 × ξ2) · (f ′(a) × g′(b) × f ′′(p−a) × g′′(q−b)) =

(ξ1 × (f ′(a) × g′(b))

)×

(ξ2 × (f ′′(p−a) × g′′(q−b))

).

Therefore, by Lemma 2.2, a + b is a breaking point of ξ · (f × g), and by (11),

ξ1 × (f ′(a) × g′(b)) = α and ξ2 × (f ′′(p−a) × g′′(q−b)) = β.

This shows that every summand in ∆(f) ∗∆(g) also occurs in ∆(f ∗ g).Now assume that we are given a shuffle ξ and a breaking point r as in (11). We have to

show that the summand α⊗ β of ∆(f ∗ g) also appears in ∆(f) ∗∆(g). Let s := p + q − r.Applying Lemma 2.5 we find numbers a, b and shuffles ξ1 ∈ Sh(a, b) and ξ2 ∈ Sh(p−a, q− b)such that a + b = r and

ξ = (ξ1 × ξ2) · (1a × ξp−a,b × 1q−b).

We first claim that a is a breaking point of f and b is a breaking point of g. Considerthe diagram of f . Suppose there is an edge connecting one of the first a bottom vertices toone of the last p − a bottom vertices. Say the former is i and the latter is j. Then in thediagram of ξ · (f × g) there is an edge connecting the bottom vertices ξ(i) and ξ(j). Butthis contradicts the fact that r is a breaking point of ξ · (f × g), since from the proof ofLemma 2.5 we know that ξ(i) ∈ {1, . . . , r} and ξ(j) ∈ {r + 1, . . . , r + s}.

Thus a ∈ B(f), and similarly b ∈ B(g). Therefore, we can write

f = (f ′(a) × f ′′(p−a)) · η−11 and g = (g′(b) × g′′(q−b)) · η−1

2 ,

where the elements in the right hand side are afforded by Lemma 2.2. We calculate:

ξ · (f × g) = (ξ1 × ξ2) · (1a × ξp−a,b × 1q−b) ·((f ′(a) × f ′′(p−a)) · η−1

1

)×

((g′(b) × g′′(q−b)) · η−1

2

)= (ξ1 × ξ2) · (1a × ξp−a,b × 1q−b) · (f ′(a) × f ′′(p−a) × g′(b) × g′′(q−b)) · (η−1

1 × η−12 )

(10)= (ξ1 × ξ2) · (f ′(a) × g′(b) × f ′′(p−a) × g′′(q−b)) · (1a × ξp−a,b × 1q−b) · (η−1

1 × η−12 )

=((

ξ1 · (f ′(a) × g′(b)))×

(ξ2 · (f ′′(p−a) × g′′(q−b))

))· (1a × ξp−a,b × 1q−b) · (η−1

1 × η−12 )

By Lemma 2.5, the permutation (1a × ξp−a,b × 1q−b) · (η−11 × η−1

2 ) is the inverse of an (a +b, p + q − a− b)-shuffle. Therefore, by the uniqueness in Lemma 2.2,

η = (η1 × η2) · (1a × ξb,p−a × 1q−b), α =(ξ1 · (f ′(a) × g′(b)), and β = ξ2 · (f ′′(p−a) × g′′(q−b)) .


This proves that the summand α⊗ β appears in ∆(f) ∗∆(g). �

Consider the following graded subspace of P :

S :=⊕n≥0

kSn .

As mentioned in the introduction, the space S carries a Hopf algebra structure, first definedby Malvenuto and Reutenauer [17]. The following result follows by comparing (6) and (8)with the definitions in [17].

Proposition 2.7. S is a Hopf subalgebra of P.

Let σ be a permutation. In the notation of [2], the element σ ∈ S corresponds to the basiselement F ∗

σ of SSym∗, or equivalently to the element Fσ−1 of SSym.

3. Hopf algebra structure of P

3.1. Inverse monoid structure and self-duality. As S, the Hopf algebra P is self-dual.To see this, recall that a block permutation is a bijection f : A → B between two setpartitions of [n]. Let f : B → A denote the inverse bijection. If f is uniform then so is f .

The diagram of f ∈ Pn is obtained by reflecting the diagram of f across a horizontal line.Note that for σ ∈ Sn ⊆ Pn we have σ = σ−1.

The operation f 7→ f is relevant to the monoid structure of Pn. Indeed, the followingproperties are satisfied

f = fff and f = ff f .

Together with (13) below, these properties imply that Pn is an inverse monoid [7, Theorem1.17]. The following properties are consequences of this fact [7, Lemma 1.18]:

fg = gf , ˜f = f

(they can also be verified directly). It follows that σ · f = f · σ−1.

The operation f 7→ f is also relevant to the Hopf algebra structure of P . Let P∗ be thegraded dual space of P :

P∗ =⊕n≥0

(kPn)∗ .

Let {f ∗ | f ∈ Pn} be the basis of (kPn)∗ dual to the basis Pn of kPn. The product on P∗ forf ∗ ∈ (kPn)∗ and g∗ ∈ (kPm)∗ is given by

f ∗ ∗ g∗ =∑

ξ∈Sh(n,m)

(f × g)∗ · ξ−1.

Let Φ : P∗ → P be the linear map such that

Φ(f ∗) := f .


Proposition 3.1. The map Φ : P∗ → P is an isomorphism of graded Hopf algebras. Inaddition, Φ∗ = Φ.

Proof. We have

Φ(f ∗ ∗ g∗) =∑

ξ∈Sh(n,m)

Φ((f × g)∗ · ξ−1

)=

∑ξ∈Sh(n,m)

ξ−1 · ˜(f × g)

=∑

ξ∈Sh(n,m)

ξ · (f × g) = Φ(f) ∗ Φ(g) .

Thus Φ preserves products. Since ˜ is an involution, Φ∗ = Φ, and hence Φ preserves coprod-ucts as well. �

3.2. Factorizable monoid structure and the weak order. Let En denote the poset ofset partitions of [n]: we say that A ≤ B if every block of B is contained in a block of A.This poset is a lattice, and this structure is related to the monoid structure of uniform blockpermutations as follows. If idA : A → A denotes the uniform block permutation which isthe identity map on the set of blocks of A, then

(13) idA · idB = idA∧B .

In other words, viewing En as a monoid under the meet operation ∧, the map

En → Pn , A 7→ idA ,

is a morphism of monoids.Any uniform block permutation f ∈ Pn decomposes (non-uniquely) as

(14) f = σ · idAfor some σ ∈ Sn and A ∈ En. Note that σ is invertible and idA is idempotent, by (13). Itfollows that Pn is a factorizable inverse monoid [6, Section 2], [16, Chapter 2.2]. Moreover,by Lemma 2.1 in [6], any invertible element in Pn belongs to Sn and any idempotent elementin Pn belongs to (the image of) En. This lemma also guarantees that in (14), the idempotentidA is uniquely determined by f (which is clear since A is the domain of f). On the otherhand, σ is not unique, and we will make a suitable choice of this factor to define a partialorder on Pn.

Consider the action of Sn on Pn by left multiplication. Given A ∈ En, the orbit of idAconsists of all uniform block permutations f : A → B with domain A, and the stabilizer isthe parabolic subgroup

SA := {σ ∈ Sn | σ(A) = A ∀A ∈ A} .

Consider the set of A-shuffles :

Sh(A) := {ξ ∈ Sn | if i < j are in the same block of A then ξ(i) < ξ(j)} .


It is well-known that these permutations form a set of representatives for the left cosets ofthe subgroup SA. Therefore, given a uniform block permutation f : A → B there is a uniqueA-shuffle ξf such that

(15) f = ξf · idA .

We use this decomposition to define a partial order on Pn as follows:

(16) f ≤ g ⇐⇒ f and g have the same domain and ξf ≤ ξg,

where the partial order on the right hand side is the left weak order on Sn (see for instance [2]).We refer to this partial order as the weak order on Pn. Thus, Pn is the disjoint union ofcertain subposets of the weak order on Sn:

Pn∼=

⊔A`[n]

Sh(A)

(in fact, each Sh(A) is a lower order ideal in Sn). Figures 4-8 show 5 of the 15 componentsof P4. Note that even when A and B are set partitions of the same type the posets Sh(A)and Sh(B) need not be isomorphic.

The partial order we have defined on Pn should not be confused with the natural partialorder which is defined on any inverse semigroup [8, Chapter 7.1], [16, Chapter 1.4].

Figure 4. The component of P4 corresponding to A = {1, 2}{3}{4}

Remark 3.2. As observed by Sloane [21], there is a connection between uniform blockpermutations and the patience games of Aldous and Diaconis [3]. A patience game is playedas follows. Start from a deck of cards numbered 1, . . . , n and arranged in any order. At eachstep, draw a card from the top of the deck and either place it on an existing pile which showsa bigger card, or start a new pile (there are thus several choices at each step). The initialdeck is a permutation of [n] and the resulting piles form a set partition of [n]. Suppose


Figure 5. The component of P4 corresponding to A = {1}{2, 3}{4}


ξ ∈ Sn. The set partitions A such that ξ ∈ Sh(A) are precisely the possible outputs ofpatience games played from a deck of cards with ξ−1(1) in the bottom, followed by ξ−1(2),up to ξ−1(n) on the top. Thus, uniform block permutations are in bijection with the pairsconsisting of the input and the output of a patience game via (ξ,A) ↔ ξ · idA.

3.3. The second basis and the Hopf algebra structure. Following the ideas of [2], weuse the weak order on Pn to define a new linear basis of the spaces kPn, on which the algebrastructure of P is simple.



Figure 8. The component of P4 corresponding to A = {1, 4}{2, 3}

For each element g ∈ Pn let

(17) Xg :=∑f≤g

f .

By Mobius inversion, the set {Xg | g ∈ Pn} is a linear basis of Pn.Note that all uniform block permutations f in the right hand side of (17) share the same

domain A ` [n], by (16).

Given set partitions A ` [p] and B ` [q], let A × B be the set partition of [p + q] whoseblocks are the blocks of A and the blocks {b + p | b ∈ B} where B is a block of B. Forexample, if A = {1, 3, 4}{2, 5}{6} ` [6] and B = {1, 4}{2}{3, 5} ` [5], then A × B ={1, 3, 4}{2, 5}{6}{7, 10}{8}{9, 11} ` [11]. This is compatible with concatenation of uniformblock permutations in the sense that if A is the (co)domain of f and B is the (co)domain ofg, then A× B is the (co)domain of f × g. In particular,

idA × idB = idA×B .

Recall the maximum (p, q)-shuffle ξp,q from (9).


Lemma 3.3. Let λ : Sh(p, q)× Sh(A)× Sh(B) −→ Sh(A× B) be defined by

λ(ξ, σ, τ) := ξ · (σ × τ) .

Endow each set of shuffles with the weak order. Then

(i) λ is bijective;(ii) λ−1 is order preserving, that is,

ξ · (σ × τ) ≤ ξ′ · (σ′ × τ ′) =⇒ ξ ≤ ξ′, σ ≤ σ′, and τ ≤ τ ′ ;

(iii) λ is order preserving when restricted to any of the following sets:

{ξp,q} × Sh(A)× Sh(B), {1p+q} × Sh(A)× Sh(B), or Sh(p, q)× {(σ, τ)} ,

for any σ ∈ Sh(A), τ ∈ Sh(B).

Proof. The proof is similar to that of Proposition 2.5 in [2]. �

The product of P takes the following simple form on the X-basis.

Proposition 3.4. Let g1 ∈ Pp1 and g2 ∈ Pp2 be uniform block permutations. Then

Xg1 ∗Xg2 = Xξp,q ·(g1×g2) .

Proof. We have:

Xg1 ∗Xg2

(17)=

∑f1≤g1f2≤g2

f1 ∗ f2

(6)=

∑ξ∈Sh(p1,p2)

∑f1≤g1f2≤g2

ξ · (f1 × f2) .

Let Ai be the domain of gi, i = 1, 2. Choose ξ ∈ Sh(p1, p2) and fi ≤ gi in Ppi, i = 1, 2.

Then Ai is the domain of fi, and by (15),

fi = ξfi· idAi

.

Henceξ · (f1 × f2) = ξ · (ξf1 × ξf2) · (idA1 × idA2) .

Since fi ≤ gi, we have ξfi≤ ξgi

, by (16). Therefore, by Lemma 3.3.(iii),

ξ · (ξf1 × ξf2) ≤ ξp,q · (ξg1 × ξg2) .

Hence, by (16),ξ · (f1 × f2) ≤ ξp,q · (g1 × g2) .

Thus every summand in Xg1 ∗Xg2 occurs in Xξp,q ·(g1×g2).Conversely, let f be a summand in Xξp,q ·(g1×g2). Since the domain of ξp,q · (g1 × g2) is

A1 ×A2, we have f = ξf · idA1×A2 . Hence

ξf ≤ ξp,q · (ξg1 × ξg2)

and from (i) and (ii) in Lemma 3.3 we obtain ξ ∈ Sh(p1, p2), σi ∈ Sh(Ai) such that

σi ≤ ξgiand ξf = ξ · (σ1 × σ2) .


Let fi := σi · idAi. Then

fi ≤ gi and f = ξ · (f1 × f2) ,

which shows that every summand in Xξp,q ·(g1×g2) occurs in Xg1 ∗Xg2 . �

From Propositions 3.4 and 3.1 we deduce:

Corollary 3.5. The Hopf algebra P is free as an algebra and cofree as a graded coalgebra.

Let V denote the space of primitive elements of P . It follows that the generating series ofP and V are related by

P(x) =1

1− V (x).

Since

P(x) = 1 + x + 3x2 + 16x3 + 131x4 + 1496x5 + 22482x6 + · · ·we deduce that

V (x) = x + 2x2 + 11x3 + 98x4 + 1202x5 + 19052x6 + · · · .

Let X∗f be the linear basis of P∗ dual to the basis Xf of P and let Mf be the image of X∗

f

under the isomorphism Φ : P∗ → P of Proposition 3.1. Explicitly, the basis Mf is uniquelydetermined by the equations

f =∑f≤g

Mg

that hold for every f ∈ P . It follows that the elements Mf , as f runs over all uniform blockpermutations that cannot be decomposed as

f = ξp,q · (f1 × f2)

with p and q > 0, form linear basis of the space of primitive elements of P . A uniform blockpermutation f ∈ Pn can be decomposed in this manner if and only if there is 0 < p < nsuch that the first p elements of the domain of f are not connected to any of the first n− pelements of the codomain. A permutation σ ∈ Sn satisfies this condition if and only if σ hasa global descent at p, in agreement with [2, Corollary 6.3].

Remark 3.6. A similar conclusion may be derived by introducing another basis

Zg :=∑g≤f

f .

This has the property that

Zg1 ∗ Zg2 = Zg1×g2 .

Note that ZidA is the element ZA introduced in (4) and so denoted throughout Section 1.3.


4. The Hopf algebra of symmetric functions in non-commuting variables

Let X be a countable set, the alphabet. A word of length n is a function w : [n] → X.Let k〈〈X〉〉 be the algebra of non-commutative power series on the set of variables X. Itselements are infinite linear combinations of words, finitely many of each length, and theproduct is concatenation of words.

The kernel of a word w of length n is the set partition K(w) of [n] whose blocks are thenon-empty fibers of w. Order the set of set partitions of [n] by refinement, as in Section 3.2.For each set partition A of [n], let

pA :=∑

K(w)≤A

w ∈ k〈〈X〉〉 .

This is the sum of all words w such that if i and j are in the same block ofA then w(i) = w(j).For instance

p{1,3}{2,4} = xyxy + xzxz + yxyx + · · ·+ x4 + y4 + z4 + · · · .

The subspace of k〈〈X〉〉 linearly spanned by the elements pA as A runs over all set parti-tions of [n], n ≥ 0, is a subalgebra Π of k〈〈X〉〉, graded by length. The elements of Π canbe characterized as those power series of finite degree that are invariant under any permu-tation of the variables. Π is the algebra of symmetric functions in non-commuting variablesintroduced by Wolf [23] and recently studied by Gebhard, Rosas, and Sagan [11, 12, 19] inconnection to Stanley’s chromatic symmetric function.

Π is in fact a graded Hopf algebra [1, 4, 5]. The coproduct is defined via evaluation ofsymmetric functions on two copies of the alphabet X. In order to describe the product andcoproduct of Π on the basis elements pA we introduce some notation.

To a set partition A ` [n] and a subset of blocks S ⊆ A we associate a new set partitionAS as follows. Write ⋃

A∈S

A = {j1, · · · , js} ⊆ [n]

with j1 < · · · < js. The set S is a partition of {j1, · · · , js}. We turn it into a partition AS of[s] by replacing each ji by i for 1 ≤ i ≤ m and preserving the block structure. For instance,if A = {1, 5}{2, 4, 6}{3, 7} and S = {1, 5}{3, 7}, then AS = {1, 3}{2, 4} ` [4].

The product and coproduct of Π are given by

pApB = pA×B ,

∆(pA) =∑

StT=A

pAS⊗ pAT

,

the sum over all decompositions of A into disjoint sets of blocks S and T . For example, ifA = {1, 2, 6}{3, 5}{4}, then

∆(pA) = pA ⊗ 1 + p{1,2,5}{3,4} ⊗ p{1} + p{1,2,4}{3} ⊗ p{1,2} + p{1,3}{2} ⊗ p{1,2,3} +

p{1,2,3} ⊗ p{1,3}{2} + p{1,2} ⊗ p{1,2,4}{3} + p{1} ⊗ p{1,2,5}{3,4} + 1⊗ pA .


Consider now the direct sum of the subspaces Zn of kPn introduced in Section 1.3:

Z :=⊕n≥0

Zn ⊂ P .

The elements ZA defined in (4) form a linear basis of Zn.

Theorem 4.1. Z is a Hopf subalgebra of P. Moreover, the map

Θ : Z → Π, Θ(ZA) := pA

is an isomorphism of graded Hopf algebras.

Proof. Recall from Remark 3.6 that ZA = ZidA , and therefore

ZA ∗ ZB = ZA×B .

Thus Θ is a morphism of algebras.To prove that Θ is a morphism of coalgebras we need to show that

∆(ZA) =∑

StT=A

ZAS⊗ ZAT

for every set partition A ` [n]. We have

ZA =∑

idA≤f

f =∑

ξ∈Sh(A)

ξ · idA

and hence

ZAS=

∑σ∈Sh(AS)

σ · idASand ZAT

=∑

τ∈Sh(AT )

τ · idAT.

Fix a decomposition A = S t T and consider the summand in ZAS⊗ ZAT

indexed byshuffles σ ∈ Sh(AS) and τ ∈ Sh(AT ). Out of this data we construct shuffles η and ξ asfollows. First we write⋃

A∈S

A = {j1, · · · , js} and⋃A∈T

A = {k1, · · · , kt}

with j1 < · · · < js and k1 < · · · < kt. We let η be the unique (s, t)-shuffle such thatη([s]) = {j1, · · · , js} (and hence η([s + 1, n]) = {k1, · · · , kt}), and we define ξ by

ξ(ji) = σ(i) ∀ i ∈ [s] and ξ(ki) = s + τ(i) ∀ i ∈ [t] .

Then ξ is increasing in each block of A, so ξ ∈ Sh(A), and by construction

ξ · idA · η = (σ · idAS)× (τ · idAT

) .

Therefore, by Lemma 2.2, s is a breaking point of ξ · idA, and by (8), (σ · idAS)⊗ (τ · idAT

)is a summand in ∆(ZA).

A similar argument shows that conversely, every summand in ∆(ZA) occurs in this manner.�


Thus the Hopf algebra of uniform block permutations P contains the Hopf algebra Π ofsymmetric functions in non-commuting variables. Note also that this reveals the existenceof a second operation on Π: according to Corollary 1.2, each homogeneous component Πn

carries an associative non-unital product that turns it into a right ideal of the monoid algebrakPn.

References

[1] Marcelo Aguiar and Swapneel Mahajan, Coxeter groups and Hopf algebras, Fields Institute Mono-graphs, Volume #23 (2006), AMS, Providence, RI.

[2] Marcelo Aguiar and Frank Sottile, Structure of the Malvenuto-Reutenauer Hopf algebra of permuta-tions, Advances in Mathematics 191 n2 (2005) 225–275.

[3] David Aldous and Persi Diaconis, Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem, Bull. Amer. Math. Soc. 36 (1999) 413–432.

[4] Nantel Bergeron, Christophe Hohlweg , Mercedes Rosas, and Mike Zabrocki, Grothendieck bialge-bras, partition lattices and symmetric functions in noncommutative variables, Electron. J. Combin. 13(2006), no. 1, Research Paper 75, 19 pp. (electronic).

[5] Nantel Bergeron and Mike Zabrocki, The Hopf algebras of symmetric functions and quasisymmetricfunctions in non-commutative variables are free and cofree, math.CO/0509265 (2005).

[6] S. Y. Chen and S. C. Hsieh, Factorizable inverse semigroups, Semigroup Forum 8 (1974) n4 283–297.[7] A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Mathematical Surveys,

No. 7 American Mathematical Society, Providence, R.I., 1961 xv+224 pp.[8] A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. II, Mathematical Surveys,

No. 7 American Mathematical Society, Providence, R.I., 1967 xv+350 pp.[9] David Easdown, James East, and D. G. FitzGerlad, Presentations of factorizable inverse monoids,

2004.[10] Desmond G. FitzGerald, A presentation for the monoid of uniform block permutations, Bull. Austral.

Math. Soc., 68 (2003) 317–324.[11] David D. Gebhard, Bruce E. Sagan, Sinks in acyclic orientations of graphs, J. Combin. Theory (B)

80 (2000) 130–146.[12] David D. Gebhard, Bruce E. Sagan, A chromatic symmetric function in noncommuting variables, J.

Alg. Combin. 13 (2001) 227–255.[13] Vaughan F. R. Jones, The Potts model and the symmetric group, in Subfactors: Proceedings of the

Tanaguchi Symposium on Operator Algebras, Kyuzeso, 1993, pp. 259-267, World Scientific, RiverEdge, NJ 1994.

[14] Masashi Kosuda, Characterization for the party algebra, Ryukyu Math. J. 13 (2000) 7–22.[15] Masashi Kosuda, Party algebra and construction of its irreducible representations, paper presented at

Formal Power Series and Algebraic Combinatorics (FPSAC01), Tempe, Arizona (USA), May 20-26,2001.

[16] Mark V. Lawson, Inverse semigroups. The theory of partial symmetries, World Scientific, River Edge,NJ, 1998. xiv+411 pp.

[17] Claudia Malvenuto and Christophe Reutenauer, Duality between quasi-symmetric functions and theSolomon descent algebra, J. Algebra 177 n3 (1995), 967–982.

[18] Paul P. Martin, Temperley-Lieb algebras for non-planar statistical mechanics - The partition algebraconstruction, J. Knot Theory and its Applicatitions, 3 n1 (1994), 51–82.

[19] Mercedes Rosas and Bruce Sagan, Symmetric functions in non-commuting variables, Trans. Amer.Math. Soc. 358 (2006), no. 1, 215–232.


[20] J.-M. Sixdeniers, K. A. Penson, and A. I. Solomon, Extended Bell and Stirling numbers from hyperge-ometric exponentiation, J. Integer Seq., 4 (2001), Article 01.1.4.

[21] Neil J. A. Sloane, An on-line version of the encyclopedia of integer se-quences, Electron. J. Combin. 1 (1994), Feature 1, approx. 5 pp. (electronic),http://akpublic.research.att.com/~njas/sequences/ol.html.

[22] Kenichiro Tanabe, On the centralizer algebra of the unitary reflection group G(m, p, n), Nagoya Math.J. 148 (1997) 113–126.

[23] M. C. Wolf, Symmetric functions of non-commuting elements, Duke Math. J. 2 (1936) 626–637.

Department of Mathematics, Texas A&M University, College Station, TX 77843, USAE-mail address: [email protected]: http://www.math.tamu.edu/∼maguiar

Department of Mathematics, Dartmouth College, Hanover, NH 03755, USAE-mail address: [email protected]: http://www.math.dartmouth.edu/∼orellana/

introduction uniform block permutation n - dartmouth …orellana/blockperm.pdf · the hopf algebra...

Documents